In a basic fiber illumination system, a light source injects light into an optical fiber and the fiber then serves to transport the injected light to a remote, desired location. At the remote location, the fiber is exposed, typically at its end surface, so that the light can escape and illuminate the remote region that is outside and near the end surface of the fiber. More recently, a fiber-based illumination system has been suggested that has multiple regions each with a different index of refraction, and this can be used to allegedly deflect the propagating light out a side of the fiber along its length. See U.S. Pat. No. 5,905,837 to Wang, et al. A 360 degree or omnidirectional deflection pattern has also been suggested, through the use of combined changes in the ratio of index of refraction between the core and the cladding, and absorption and scattering coefficients in the fiber. See U.S. Pat. No. 6,714,711 to Lieberman, et al. A subsequent effort has suggested the use of a blazed diffraction grating in the core of the fiber, for diffracting the light out of the fiber, and a convex lens structure outside of the fiber is to receive the diffracted light and establish a linear illumination field. See U.S. Pat. No. 7,437,035 to Carver, et al. However, the efficacy of such techniques in efficiently producing illumination light having broadband visible content (also referred to here as white light) while ensuring its uniformity along the length of the fiber is not apparent.